Abstract
The correct solvability of initial-value problems for integrodifferential equations with unbounded operator coefficients in Hilbert spaces is determined. Numerous problems of hereditary mechanics and thermal physics have motivated the study of such equations. Models taking into account a Kelvin–Voight friction are considered. A spectral analysis of the operator functions, which are the symbols of integro-differential equations considered in this paper, is made.
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T. Ya. Azizov, N. D. Kopachevsky, and L. D. Orlova, “An operator approach to the study of the Oldroyd hydrodynamic model,” Math. Notes., 65, No. 6, 773–776 (1999).
C. M. Dafermos, “Asymptotic stability in viscoelasticity,” Arch. Ration. Mech. Anal., 37, 297–308 (1970).
W. Desch, “Exponential stabilization of Volterra integrodifferential equations in Hilbert space,” J. Differ. Equ., 70, 366–389 (1987).
A. Eremenko and S. Ivanov, “Spectra of the Gurtin–Pipkin type equations,” SIAM J. Math. Anal., 43, 2296–2306 (2011).
R. O. Griniv and A. A. Shkalikov, “On an operator pencil arising in the problem of beam oscillation with internal damping,” Math. Notes, 56, No. 2, 840–851 (1994).
M. E. Gurtin and A. C. Pipkin, “Theory of heat conduction with finite wave speed,” Arch. Ration. Mech. Anal., 31, 113–126 (1968).
S. Ivanov and L. Pandolfi, “Heat equations with memory: lack of controllability to the rest,” J. Math. Anal. Appl., 355, 1–11 (2009).
J. L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer-Verlag, Berlin–Heidelberg–New York (1972).
A. V. Lykov, Problems of Heat and Mass Transfer [in Russian], Nauka i Technika, Minsk (1976).
D. A. Medvedev and V. V. Vlasov, “Functional-differential equations in Sobolev spaces and related problems in spectral theory,” J. Math. Sci., 164, No. 5, 653–841 (2008).
A. I. Miloslavsky, “On the stability of some classes of evolution equations,” Sib. Mat. Zh., 26, 118–132 (1985).
A. I. Miloslavsky, “Spectral properties of an operator pencil arising in viscoelasticity,” Deposited in Ukr. NIINTI, No. 1229-UK87, Kharkov (1987).
A. I. Miloslavsky, “Instability spectrum of an operator bundle,” Math. Notes, 49, No. 4, 391–395 (1991).
J. E. Muñoz Rivera, M. G. Naso and F. M. Vegni, “Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,” J. Math. Anal. Appl., 286, 692–704 (2003).
V. V. Palin and E. V. Radkevich, “Conservation laws and their hyperbolic regularizations,” J. Math. Sci., 164, No. 6, 922–944 (2010).
L. Pandolfi, “The controllability of the Gurtin–Pipkin equations: a cosine operator approach,” Appl. Math. Optim., 52, 143–165 (2005).
J. Prüss, Evolutionary Integral Equations amd Applications, Birkhauser, Basel–Boston–Berlin (1993).
N. A. Rautian, A. S. Shamaev, and V. V. Vlasov, “Solvability and spectral analysis of integrodifferential equations arising in the theory of heat transfer and acoustics,” Dokl. Math., 82, No. 2, 684–687 (2010).
N. A. Rautian, A. S. Shamaev and V. V. Vlasov, “Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics,” J. Math. Sci., 190, No. 1, 34–65 (2013).
N. A. Rautian and V. V. Vlasov, “Analysis of integrodifferential equations arising in the theory of viscoelasticity,” Izv. VUZ. Matematika, 6, 56–60 (2012).
E. Sanches Palensia, Nonhomogeneous Media and Vibration Theory, Springer, Heidelberg (1980).
A. S. Shamaev and V. V. Shumilova, “Averaging the acoustics equations for a viscoelastic material with channels filled with a viscous compressible fluid,” Fluid Dyn., 46, No. 2, 250–261 (2011).
V. V. Vlasov, “On the solvability and properties of solutions of functional-differential equations in Hilbert spaces,” Sb. Math., 186, No. 8, 1147–1172 (1995).
V. V. Vlasov and N. A. Rautian, “Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential quaitons,” J. Math. Sci., 179, No. 3, 390–414 (2011).
V. V. Vlasov and J. Wu, “Spectral analysis and solvability of abstract hyperbolic equations with aftereffect,” Differ. Equ., 45, No. 4, 539–548 (2009).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 45, Proceedings of the Sixth International Conference on Differential and Functional Differential Equations and International Workshop “Spatio-Temporal Dynamical Systems” (Moscow, Russia, 14–21 August, 2011). Part 1, 2012.
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Vlasov, V.V., Rautian, N.A. & Shamaev, A.S. Analysis of Operator Models Arising in Problems of Hereditary Mechanics. J Math Sci 201, 673–692 (2014). https://doi.org/10.1007/s10958-014-2019-4
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DOI: https://doi.org/10.1007/s10958-014-2019-4